NOTE: this is a working draft of the questions we would use to assist students in making a valid distinction between induction and deduction. Please feel free to leave us notes, but please do not delete what we have put here. Your input and the process are very valuable so if you would add your thoughts in a different color, we would be very HAPPY!!
"bansho keikaku" (blackboard planning: only a piece - not the whole pi) We are imagining that the teacher would be noting answers on a two column type note taking without headings. These heading should be put in at the end or after the
Questions, Round 2
IDEA:
Two columns: filled with answers to the following questions. Teacher led discussion. If the student's do not respond, the teacher will look at the posters around the room and try to help them remember by mentioning key words and asking questions. The teacher will setup two columns on the board and will prompt: On one side of the index card put down what you were thinking about while you were exploring and what you were thinking about while proving. After private think time, the class will brainstorm the blue questions together.
These are written on the board and key inductive/deductive words are circled.
Thinking while “exploring”
Thinking while “proving”
What was the starting point in your thinking about this problem? Anticipated Responses: I played with tiles, tried to fill a piece of paper with tiles. I drew shapes. I used the envelopes. You asked me to fill a piece of paper with tiles. I wrote down what I saw when I tried to tile. I tried to make the pictures you showed us. I tried to make something that looked nice.
How did you determine that you were “done” with a figure?
Anticipated Response: No gaps. The figure could not fit together. The tiles fit together with nice gaps in the middle. I knew the angles have to work.
What did you think you knew or didn't know after you played with the shapes? (ie. This worked, this will always work, this won’t work, . . .)
Anticipated Response: I knew that a triangle worked. I knew that a hexagon worked. I knew a heptagon didn't work.
What did you wonder about?
Anticipated Response: Why did the ones that tile work? Why didn't 5 work? Is there a pattern? Why did you only give me these envelopes?
What was the starting point the proving of this problem? Anticipated Responses: I started to think about the numbers. I stopped playing. I thought about the properties of regular polygons.
What information moved us forward in the proof? Anticipated Responses: The common vertex, the 360 degrees, the definition of a polygon. Using interior angles. Using factor pairs. The possible interior angle values of a polygon.
How did we know that we were done with the proof? Anticipated Response: We had good reasons for why some worked and some didnt. Because I actually had the polygon tilings. I knew that only three worked. I started thinking about the angles and used the sizes to conclude what they had to be. The proof explained why some of the one's (not) in the envelopes didn't tile.
How did we decide that we could conclude that only ,_, and _ would tessellate the plane?
What words, ideas, processes do we notice are the same in column 1? . . . Column 2?
Add to the columns from vignettes
Also like thinking while playing
Also like thinking while proving
discussion leads students to really see the difference (. . . something, something)
Two columns:
inductive
deductive
We need to label these types of thinking....
Possible questions:
What did you think that you knew after you played with the shapes?
How certain were you?
What do you think you know now?
How certain are you now?
What made the difference between then and now?
What were the differences between the way you thought about the question when you were playing and when you were trying to prove it?
What is the difference between an answer that is reasonable, and an answer that must be true?
What methods do we use to come up with a reasonable answer? An answer that must be?
What’s the difference in level of certainty?
An overriding question: What is the difference between truth in science, English, society versus truth in mathematics?
QUESTIONS / CONCERNS :
Uncomfortable with the word "truth..." it's not a mathematical word.... "True" means a logical consequence from aa system of definitions and axioms and postulates
Perhaps change "played" to investigated?
Not sure how the type of thinking, inductive vs. deductive, are brought out through the questions.
We were thinking of using the board to write down the students answers as they fall under the two categories "inductive and deductive". However, even with this, these questions need some help. - Stacey
You might try some "why?" questions as well, e.g. how certain are you of your result? Why?
NOTE: this is a working draft of the questions we would use to assist students in making a valid distinction between induction and deduction. Please feel free to leave us notes, but please do not delete what we have put here. Your input and the process are very valuable so if you would add your thoughts in a different color, we would be very HAPPY!!
"bansho keikaku" (blackboard planning: only a piece - not the whole pi)
We are imagining that the teacher would be noting answers on a two column type note taking without headings. These heading should be put in at the end or after the
Questions, Round 2
IDEA:
Two columns: filled with answers to the following questions. Teacher led discussion. If the student's do not respond, the teacher will look at the posters around the room and try to help them remember by mentioning key words and asking questions. The teacher will setup two columns on the board and will prompt: On one side of the index card put down what you were thinking about while you were exploring and what you were thinking about while proving. After private think time, the class will brainstorm the blue questions together.
These are written on the board and key inductive/deductive words are circled.
Anticipated Responses: I played with tiles, tried to fill a piece of paper with tiles. I drew shapes. I used the envelopes. You asked me to fill a piece of paper with tiles. I wrote down what I saw when I tried to tile. I tried to make the pictures you showed us. I tried to make something that looked nice.
How did you determine that you were “done” with a figure?
Anticipated Response: No gaps. The figure could not fit together. The tiles fit together with nice gaps in the middle. I knew the angles have to work.
What did you think you knew or didn't know after you played with the shapes? (ie. This worked, this will always work, this won’t work, . . .)
Anticipated Response: I knew that a triangle worked. I knew that a hexagon worked. I knew a heptagon didn't work.
What did you wonder about?
Anticipated Response: Why did the ones that tile work? Why didn't 5 work? Is there a pattern? Why did you only give me these envelopes?
Anticipated Responses: I started to think about the numbers. I stopped playing. I thought about the properties of regular polygons.
What information moved us forward in the proof?
Anticipated Responses: The common vertex, the 360 degrees, the definition of a polygon. Using interior angles. Using factor pairs. The possible interior angle values of a polygon.
How did we know that we were done with the proof?
Anticipated Response: We had good reasons for why some worked and some didnt. Because I actually had the polygon tilings. I knew that only three worked. I started thinking about the angles and used the sizes to conclude what they had to be. The proof explained why some of the one's (not) in the envelopes didn't tile.
How did we decide that we could conclude that only ,_, and _ would tessellate the plane?
Add to the columns from vignettes
discussion leads students to really see the difference (. . . something, something)
Two columns:
We need to label these types of thinking....
Possible questions:
What did you think that you knew after you played with the shapes?
How certain were you?
What do you think you know now?
How certain are you now?
What made the difference between then and now?
What were the differences between the way you thought about the question when you were playing and when you were trying to prove it?
What is the difference between an answer that is reasonable, and an answer that must be true?
What methods do we use to come up with a reasonable answer? An answer that must be?
What’s the difference in level of certainty?
An overriding question:
What is the difference between truth in science, English, society versus truth in mathematics?
QUESTIONS / CONCERNS :
Uncomfortable with the word "truth..." it's not a mathematical word.... "True" means a logical consequence from aa system of definitions and axioms and postulates
Perhaps change "played" to investigated?
Not sure how the type of thinking, inductive vs. deductive, are brought out through the questions.
We were thinking of using the board to write down the students answers as they fall under the two categories "inductive and deductive". However, even with this, these questions need some help. - Stacey
You might try some "why?" questions as well, e.g. how certain are you of your result? Why?